Implementation steps and time complexity analysis of radix sorting algorithm in PHP.

Implementation steps and time complexity analysis of radix sorting algorithm in PHP

Radix Sort is a commonly used linear time complexity (O(n )) sorting algorithm achieves sorting by comparing and assigning elements bit by bit. In this article, we will introduce the implementation steps of the radix sort algorithm and analyze its time complexity.

The basic idea of ​​radix sorting is to allocate all elements to be compared (positive integers) to a limited number of buckets, and then collect the elements in each bucket in turn to finally complete the sorting.

The implementation steps are as follows:

1. Initialize the bucket array: Create a two-dimensional array to represent the buckets. Each bucket is a one-dimensional array. The number of buckets is determined by the maximum number of digits of elements to be sorted.
2. Find the maximum number of digits: Traverse the array to be sorted, find the largest element, and determine its number of digits.
3. Distribute and collect bit by bit: from low bit to high bit, take out the value of the corresponding number of digits of each element in turn, and allocate the elements to the corresponding buckets. Then collect them back to the original array in the order of the buckets.
4. Repeat step 3: allocate and collect the high bits in sequence until the highest bit is allocated.
5. Complete sorting: After multiple allocations and collections, the array to be sorted has become ordered.

The following is an example of PHP code for radix sorting:

function radixSort(array $arr): array {
    // 找到待排序数组的最大值
    $max = max($arr);
    
    // 确定最大值的位数
    $maxDigit = strlen((string)$max);
    
    // 初始化桶数组
    $buckets = [];
    for ($i = 0; $i < 10; $i++) {
        $buckets[$i] = [];
    }
    
    // 依次按位进行分配和收集
    for ($digit = 1; $digit <= $maxDigit; $digit++) {
        // 分配到桶中
        foreach ($arr as $num) {
            $index = ($num / pow(10, $digit - 1)) % 10;
            array_push($buckets[$index], $num);
        }
        
        // 按照桶的顺序进行收集
        $pos = 0;
        for ($i = 0; $i < 10; $i++) {
            while (!empty($buckets[$i])) {
                $arr[$pos] = array_shift($buckets[$i]);
                $pos++;
            }
        }
    }
    
    return $arr;
}

// 测试
$arr = [170, 45, 75, 90, 802, 24, 2, 66];
$result = radixSort($arr);
print_r($result);

Time complexity analysis:

• If the number of digits of the elements to be sorted is d, the bucket The number of is k, then the time complexity of radix sorting is O(d * (n k)).
• In the worst case, the number of digits of elements to be sorted is equal to the number of buckets, that is, d = k. At this time, the time complexity is O(2 * n).
• Under average circumstances, the number of digits of elements to be sorted has nothing to do with the number of buckets, that is, d

Although radix sorting can achieve linear time complexity, its space complexity is high and additional bucket arrays are needed to store elements. Additionally, in the case of dealing with negative numbers, conversion and inversion operations are required on the elements. However, in practical applications, if the data to be sorted is small or the environment has sufficient memory, radix sorting is still an efficient sorting algorithm.

To sum up, this article introduces the implementation steps of the radix sorting algorithm in PHP and analyzes its time complexity. By comparing and assigning elements bit by bit, radix sort can complete the sorting task efficiently. When writing practical applications, you can choose an appropriate sorting algorithm based on the characteristics of the elements to be sorted to improve performance.

The above is the detailed content of Implementation steps and time complexity analysis of radix sorting algorithm in PHP.. For more information, please follow other related articles on the PHP Chinese website!